A308412 Indices of Gaussian primes on a square spiral.

3, 5, 7, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 54, 60, 62, 68, 70, 76, 78, 82, 84, 88, 90, 92, 94, 98, 100, 102, 104, 108, 110, 112, 114, 118, 120, 122, 126, 128, 132, 134, 138, 140, 144, 146, 150, 152, 156, 158

Offset

1,1

Comments

These are the numbers k > 0 such that A174344(k) + i*A274923(k) is a Gaussian prime (where i denotes the imaginary unit).

For symmetry reasons, we obtain the same sequence when considering a clockwise or a counterclockwise square spiral, or when initially moving towards any unit direction.

All terms except the first four are even.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A308412

Eric Weisstein's World of Mathematics, Gaussian Prime

Example

The first terms displayed on the center of a counterclockwise square spiral are:
  y\x|    -5   -4   -3   -2   -1    0   +1   +2   +3   +4   +5
  ---+--------------------------------------------------------
   +5|     *--100----*---98----*----*----*---94----*---92----*
     |     |                                                 |
   +4|   102    *----*----*---62----*---60----*----*----*   90
     |     |    |                                       |    |
   +3|     *    *    *---36----*---34----*---32----*    *    *
     |     |    |    |                             |    |    |
   +2|   104    *   38    *---16----*---14----*   30    *   88
     |     |    |    |    |                   |    |    |    |
   +1|     *   68    *   18    5----*----3   12    *   54    *
     |     |    |    |    |    |         |    |    |    |    |
    0|     *    *   40    *    *    *----*    *   28    *    *
     |     |    |    |    |    |              |    |    |    |
   -1|     *   70    *   20    7----*----9---10    *   52    *
     |     |    |    |    |                        |    |    |
   -2|   108    *   42    *---22----*---24----*---26    *   84
     |     |    |    |                                  |    |
   -3|     *    *    *---44----*---46----*---48----*----*    *
     |     |    |                                            |
    4|   110    *----*----*---76----*---78----*----*----*---82
     |     |
    5|     *--112----*--114----*----*----*--118----*--120----*

Maple

SP:= proc(n) option remember; local k;
k:=floor(sqrt(4*n-7)) mod 4;
procname(n-1) -I*exp(I*k*Pi/2)
end proc:
SP(1):= 0:
select(i -> GaussInt:-GIprime(SP(i)), [$1..1000]); # Robert Israel, May 20 2024

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A174344, A274923.

Sequence in context: A214315 A249098 A287774 * A327254 A054684 A294601

Adjacent sequences: A308409 A308410 A308411 * A308413 A308414 A308415

Keyword

nonn,changed

Author

Rémy Sigrist, Jun 01 2019

Status

approved